The European airlines data set consists of \(N=450\)  nodes that represent European airports
and has \(L=37\)  layers that represent different airlines operating in Europe. There are
3588 edges, which represent available routs. This network can be represented by a
fourth-order adjacency tensor \(\mathcal {A}\in \mathbb {R}^{N\times L\times N\times L}\)  such that
\(\mathcal {A}(i,\ell ,j,\ell )=1\)  if there is a flight connecting airports \(i\)  and \(j\)  with airline
\(\ell \) . Moreover, \(\mathcal {A}(i,\ell ,i,k)=1\)  for every \(1\le \ell ,k\le L\)  to reflect the effort
required to change airlines for connecting flights. The network can be downloaded
from [1]}. Similarly as Taylor et al. [2]}, we only include \(N=417\) 
nodes from the largest connected component of the network. We compute the multilayer total
communicability of the nodes and the multilayer Katz centrality using Algorithm
REF  to approximate the tensor exponential and the tensor resolvent functions.
Table REF  lists the top 10 central nodes. We obtained similar ranking as
reported in [3]}, where the authors applied Krylov subspace methods to the
supra-adjacency matrix of the network in order to compute matrix function-based centrality
measures such as the Katz centrality.
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